EE3CL4: Introduction to Linear Control Systems

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1 1 / 17 EE3CL4: Introduction to Linear Control Systems Section 7: McMaster University Winter 2018

2 2 / 17 Outline 1

3 4 / 17 Cascade compensation Throughout this lecture we consider the case of H(s) = 1. We have looked at using lead compensators to improve the transient performance of a closed loop lag compensators to improve the steady state error responses without changing the closed loop transient response too much. What if we wanted to do both?

4 5 / 17 Lead-lag compensation Apply lead design techniques to G(s) to adjust the closed loop transient response Then apply lag design techniques to G C,lead (s)g(s) to improve steady state error response without changing the closed loop transient response too much Resulting compensator: G C (s) = G C,lag (s)g C,lead (s) = K C,lagK C,lead (s + z lag )(s + z lead ) (s + p lag )(s + p lead ) How can we gain insight into what the compensator is doing?

5 6 / 17 Lead-lag approximation G C,lag (s)g C,lead (s) = K C,lagK C,lead (s + z lag )(s + z lead ) (s + p lag )(s + p lead ) Recall that for frequencies between z lead and p lead, lead compensator acts like a differentiator for frequencies between p lag and z lag, lag compensator acts like an integrator Rewrite: G C,lag (s)g C,lead (s) = K C,ll (s + z lag )(s + z lead ) (s + p lag )(1 + s/p lead ) as p lead gets big, and p lag gets small this starts to look like G C,lag (s)g C,lead (s) K C,ll (s + z lag )(s + z lead ) s for the values of s that are of greatest interest. Not physically realizable (more zeros than poles), but helpful approximation

6 7 / 17 Lead-lag to PID G C,lag (s)g C,lead (s) K C,ll (s + z lag )(s + z lead ) s Do a partial fraction on RHS and you get G C,lag (s)g C,lead (s) K P + K I s + K Ds With H(s) = 1, input to the compensator is e(t) = r(t) y(t) Compensator output: u(t) = L 1{ G C,lag (s)g C,lead (s)e(s) } = u(t) K P e(t) + K I That is, (approximately) PID control e(t) dt + K D de(t) dt

7 8 / 17 Variants of PID control PID: G c (s) = K P + K I /s + K D s. With K D = 0 we have a PI controller, G PI (s) = ˆK P + ˆK I /s. With K I = 0 we have a PD controller, G PD (s) = K P + K D s. As implicit in our derivation, a PID controller can be realized as the cascade of a PI controller and a PD controller; i.e., G PI (s)g PD (s) can be written as K P + K I /s + K D s

8 9 / 17 PID control and root locus Transfer function of idealized PID controller: G C (s) = K P + K I /s + K D s = K D(s + z 1 )(s + z 2 ) s That is, controller adds two zeros and a pole to the open loop transfer function The pole is at the origin The zeros can be arbitrary real numbers, or an arbitrary complex conjugate pair This provides considerable flexibility in re-shaping the root locus

9 10 / 17 PID Tuning with G c (s) = K P + K I /s + K D s. How should we choose K P, K I and K D? Can formulate as a optimization problem; e.g., Find K P, K I and K D that minimize the settling time, subject to the damping ratio being greater than ζ min, the position and velocity error constants being greater than K posn,min and K v,min, the error constant for a step disturbance being greater than K dist,posn,min, and the loop being stable Typically difficult to find the optimal solution Many ad-hoc techniques that usually find good solutions have been proposed.

10 11 / 17 Zeigler Nichols Tuning Two well established methods for finding a good solution in some common scenarios Often useful in practice because they can be applied to cases in which the model has to be measured (no analytic transfer function) We will look at the ultimate gain method This is based on the step response of the system However, the method is only suitable for a certain class of systems and a certain class of design goals You need to make sure that the system you wish to control falls into an appropriate class. You also need to ensure that the ZN tuning goals match your design goals. The ZN tuning scheme gives considerable weight to the response to disturbances

11 12 / 17 Ultimate Gain Zeigler Nichols Tuning 1 Set K I and K D to zero. 2 Increase K P until the system is marginally stable (Poles on the jω-axis) 3 The value of this gain is the ultimate gain, K U 4 The period of the sustained oscillations is called the ultimate period, T U (or P U ). (The position of the poles on the jω-axis is 2π/T U ) 5 The gains are then chosen using the following table

12 13 / 17 Ultimate Gain Zeigler Nichols Tuning

13 14 / 17 Manual refinement One way in which the design can be improved, is searching for nearby gains that improve the performance The following table provides guidelines for that local search. These are appropriate for a broad class of systems

14 15 / 17 Example G(s) = 1 s(s+b)(s+2ζω n), with b = 10, ζ = 1/ 2 and ω n = 4. Step 2: Plot root locus of G(s) to find K U and T U Step 3: K U = 885.5, Step 4: marginally stable poles: ±j7.5; T U = 0.83s Step 5: K P = 521.3, K I = , K D = 55.1

15 16 / 17 Example Step response of ZN tuned closed loop, K P = 521.3, K I = , K D = 55.1

16 17 / 17 Example Step response with manually modified gains, K P = 370, K I = 100, K D = 60

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